Question: At the end of the year, the Math Club decided to hold an election for which 5 equal officer positions were available. However, 16 candidates were nominated, of whom 7 were past officers. Of all possible elections of the officers, how many will have at least 1 of the past officers?
Solution: The number of total ways to choose the 5 officers is $\binom{16}{5} = 4368$.  Of these, the number of ways to choose the officers without ANY of the past officers is $\binom{9}{5} = 126$. Thus, the number of ways to choose the 5 officers with at least 1 past officer is $4368 - 126 = \boxed{4242}.$